Comparison theorem

1. The problem statement, all variables and given/known data
Use the Comparison Theorem to determine whether the integral below is convergent or divergent:

e^-x / sqrt x dx integrated from 0-->1

3. The attempt at a solution
I think it is convergent because 1/e^x is convergent. I set the original integral less than or equal to the integral of 1/(e^x) dx

When I solved for it, I got -1/e + 1, therefore it is convergent. However, my professor marked my paper as saying it's not true. He set the original integral less than or equal to 1/sqrt x, and solving for that, got 2. Why can't my comparison hold true?